This construction is known as the “square ice” model, because in statistical mechanics it can be used to model ice (as in normal physical ice, made from water). The vertices correspond to oxygen atoms, each of which is bonded to four hydrogen atoms that lie “within” the bonds of the lattice (and each hydrogen atom is thus bonded to two oxygen atoms). The hydrogen atoms are not positioned symmetrically along the bonds, but are closer to one of their oxygen atoms than the other one; the arrow on each bond of the lattice points to the closest oxygen atom for that bond’s hydrogen atom.

OK, here’s the pretty picture, i.e. the embroidered loop diagram corresponding to the permutation matrix [16, 15, 13, 12, 19, 22, 9, 23, 26, 8, 6, 29, 30, 5, 31, 32, 1, 2, 28, 3, 4, 27, 25, 7, 10, 24, 11, 14, 21, 20, 18, 17]. The small white stitches mark the non-zero matrix entries where the arrows on the lattice change direction and hence the corresponding stitch is twice as long as usual.

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ALT

Further thoughts on the pretty picture. While I was stitching it, I was wondering if there were any other permutation matrices that would produce Fibonacci snowflakes. There are indeed — after I finished it, I turned it over and there was the order 3 snowflake in the centre. (The photo below is less clear than the front view because of the knots where I had to join thread, but hopefully it’s clear enough.) This corresponds to taking our original permutation matrix “backwards”, i.e. [17, 18, 20, 21, 14, 11, 24, 10, etc].

I like this because it’s a result that’s trivial to see once you’ve stitched it but is not at all obvious from any other way of rendering it (e.g. drawing it on paper or digitally).

#MathArt #MathsArt #Maths #Math #Embroidery #Sashiko

ALT

@geoffl Oh, very nice! So now we have two ways of making the result obvious :)

I had been pondering writing a renderer in order to make it easier to play around with ideas for finding matrices for the higher-order snowflakes. I have a hunch that the matrices will need to have fourfold symmetry like this one does. I suppose really though the first step should be emailing Sébastien Labbé to find out how they discovered this one.

@kake I'd programned things in BASIC to generate Hitomezashi stitch patterns before and was interested in this variation when I saw you post about it. As it has 4 fold symetry you really only need 1/4 of the probability matrix to genarate it. Unlike the normal matrix where a location only excludes any others in the same row or column this requires each cell chosen in the complete matrix to also have 3 others, as 90° rotations, also selected.

@geoffl Indeed, yes, that’s what I meant by my hunch. However: the snowflakes all have fourfold symmetry but I don’t think it *necessarily* follows that the matrices also have to have it. It might be possible to construct a non-symmetric matrix that has a symmetric snowflake in the middle.

@kake With a bigger grid to stitch across you can have a matrix that has larger offsets on one side than the other without breaking the snowflake (as no points of the probability matrix lay on, or need to lay next to, the boundry of the snowflake) but I can't tell if you can fill the rest of the probability matrix without breaking it. Feels like it should work.

@kake Higher order Fibonacci snowflake.

It's a 4th order snowflake generated using a 68x68 permutation matrix with rotational symmetry.

ALT